The first solvable group that is not polycyclic is $\mathbb{Z}[1/2]\rtimes \mathbb{Z}$ (where the automorphism is given by multiplication with 2) and I do not see a way of embedding it into $GL_n(\mathbb{Z})$ for some $n$.

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It is a theorem of Mal'cev that all solvable subgroups of $GL(n,\mathbb Z)$ are polycyclic, and a theorem of Auslander that every polycyclic group is isomorphically embeddable in $GL(n,\mathbb Z)$, for some $n$. Auslander's theorem was later reproved by Swan purely algebraically by adapting the proof of Ado's theorem.